共查询到20条相似文献,搜索用时 46 毫秒
1.
An Efficient Positivity-Preserving Finite Volume Scheme for the Nonequilibrium Three-Temperature Radiation Diffusion Equations on Polygonal Meshes 下载免费PDF全文
This paper develops an efficient positivity-preserving finite volume scheme
for the two-dimensional nonequilibrium three-temperature radiation diffusion equations on general polygonal meshes. The scheme is formed as a predictor-corrector algorithm. The corrector phase obtains the cell-centered solutions on the primary mesh,
while the predictor phase determines the cell-vertex solutions on the dual mesh independently. Moreover, the flux on the primary edge is approximated with a fixed
stencil and the nonnegative cell-vertex solutions are not reconstructed. Theoretically,
our scheme does not require any nonlinear iteration for the linear problems, and can
call the fast nonlinear solver (e.g. Newton method) for the nonlinear problems. The
positivity, existence and uniqueness of the cell-centered solutions obtained on the corrector phase are analyzed, and the scheme on quasi-uniform meshes is proved to be $L^2$- and $H^1$-stable under some assumptions. Numerical experiments demonstrate the
accuracy, efficiency and positivity of the scheme on various distorted meshes. 相似文献
2.
Jinjing Xu Fei Zhao Zhiqiang Sheng & Guangwei Yuan 《Communications In Computational Physics》2021,29(3):747-766
In this paper we propose a new nonlinear cell-centered finite volume scheme
on general polygonal meshes for two dimensional anisotropic diffusion problems,
which preserves discrete maximum principle (DMP). The scheme is based on the so-called diamond scheme with a nonlinear treatment on its tangential flux to obtain a
local maximum principle (LMP) structure. It is well-known that existing DMP preserving diffusion schemes suffer from the fact that auxiliary unknowns should be presented as a convex combination of primary unknowns. In this paper, to get rid of
this constraint a nonlinearization strategy is introduced and it requires only a second-order accurate approximation for auxiliary unknowns. Numerical results show that
this scheme has second-order accuracy, preserves maximum and minimum for solutions and is conservative. 相似文献
3.
A Constrained Finite Element Method Based on Domain Decomposition Satisfying the Discrete Maximum Principle for Diffusion Problems 下载免费PDF全文
In this paper, we are concerned with the constrained finite element method
based on domain decomposition satisfying the discrete maximum principle for diffusion
problems with discontinuous coefficients on distorted meshes. The basic idea of
domain decomposition methods is used to deal with the discontinuous coefficients. To
get the information on the interface, we generalize the traditional Neumann-Neumann
method to the discontinuous diffusion tensors case. Then, the constrained finite element
method is used in each subdomain. Comparing with the method of using the
constrained finite element method on the global domain, the numerical experiments
show that not only the convergence order is improved, but also the nonlinear iteration
time is reduced remarkably in our method. 相似文献
4.
A New Interpolation for Auxiliary Unknowns of the Monotone Finite Volume Scheme for 3D Diffusion Equations 下载免费PDF全文
Fei Zhao Xiang Lai Guangwei Yuan & Zhiqiang Sheng 《Communications In Computational Physics》2020,27(4):1201-1233
A monotone cell-centered finite volume scheme for diffusion equations on
tetrahedral meshes is established in this paper, which deals with tensor diffusion coefficients and strong discontinuous diffusion coefficients. The first novelty here is to
propose a new method of interpolating vertex unknowns (auxiliary unknowns) with
cell-centered unknowns (primary unknowns), in which a sufficient condition is given
to guarantee the non-negativity of vertex unknowns. The second novelty of this paper
is to devise a modified Anderson acceleration, which is based on an iterative combination of vertex unknowns and will be denoted as AA-Vertex algorithm, in order to solve
the nonlinear scheme efficiently. Numerical testes indicate that our new method can
obtain almost second order accuracy and is more accurate than some existing methods.
Furthermore, with the same accuracy, the modified Anderson acceleration is much
more efficient than the usual one. 相似文献
5.
High-Order Positivity-Preserving Well-Balanced Discontinuous Galerkin Methods for Euler Equations with Gravitation on Unstructured Meshes 下载免费PDF全文
Weijie Zhang Yulong Xing Yinhua Xia & Yan Xu 《Communications In Computational Physics》2022,31(3):771-815
In this paper, we propose a high-order accurate discontinuous Galerkin
(DG) method for the compressible Euler equations under gravitational fields on unstructured meshes. The scheme preserves a general hydrostatic equilibrium state and
provably guarantees the positivity of density and pressure at the same time. Comparing with the work on the well-balanced scheme for Euler equations with gravitation
on rectangular meshes, the extension to triangular meshes is conceptually plausible
but highly nontrivial. We first introduce a special way to recover the equilibrium state
and then design a group of novel variables at the interface of two adjacent cells, which
plays an important role in the well-balanced and positivity-preserving properties. One
main challenge is that the well-balanced schemes may not have the weak positivity
property. In order to achieve the well-balanced and positivity-preserving properties
simultaneously while maintaining high-order accuracy, we carefully design DG spatial discretization with well-balanced numerical fluxes and suitable source term approximation. For the ideal gas, we prove that the resulting well-balanced scheme, coupled with strong stability preserving time discretizations, satisfies a weak positivity
property. A simple existing limiter can be applied to enforce the positivity-preserving
property, without losing high-order accuracy and conservation. Extensive one- and
two-dimensional numerical examples demonstrate the desired properties of the proposed scheme, as well as its high resolution and robustness. 相似文献
6.
Boyang Yu Hongtao Yang Yonghai Li & Guangwei Yuan 《Communications In Computational Physics》2022,31(5):1489-1524
We apply the monotonicity correction to the finite element method for the
anisotropic diffusion problems, including linear and quadratic finite elements on triangular meshes. When formulating the finite element schemes, we need to calculate
the integrals on every triangular element, whose results are the linear combination
of the two-point pairs. Then we decompose the integral results into the main and
remaining parts according to coefficient signs of two-point pairs. We apply the nonlinear correction to the positive remaining parts and move the negative remaining parts
to the right side of the finite element equations. Finally, the original stiffness matrix
can be transformed into a nonlinear M-matrix, and the corrected schemes have the
positivity-preserving property. We also give the monotonicity correction to the time
derivative term for the time-dependent problems. Numerical experiments show that
the corrected finite element method has monotonicity and maintains the convergence
order of the original schemes in $H^1$-norm and $L^2$-norm, respectively. 相似文献
7.
A Decoupled and Positivity-Preserving DDFVS Scheme for Diffusion Problems on Polyhedral Meshes 下载免费PDF全文
We propose a decoupled and positivity-preserving discrete duality finite
volume (DDFV) scheme for anisotropic diffusion problems on polyhedral meshes with
star-shaped cells and planar faces. Under the generalized DDFV framework, two sets
of finite volume (FV) equations are respectively constructed on the dual and primary
meshes, where the ones on the dual mesh are derived from the ingenious combination
of a geometric relationship with the construction of the cell matrix. The resulting system on the dual mesh is symmetric and positive definite, while the one on the primary
mesh possesses an M-matrix structure. To guarantee the positivity of the two categories of unknowns, a cutoff technique is introduced. As for the local conservation, it
is conditionally maintained on the dual mesh while strictly preserved on the primary
mesh. More interesting is that the FV equations on the dual mesh can be solved independently, so that the two sets of FV equations are decoupled. As a result, no nonlinear
iteration is required for linear problems and a general nonlinear solver could be used
for nonlinear problems. In addition, we analyze the well-posedness of numerical solutions for linear problems. The properties of the presented scheme are examined by
numerical experiments. The efficiency of the Newton method is also demonstrated by
comparison with those of the fixed-point iteration method and its Anderson acceleration. 相似文献
8.
Continuous Finite Element Subgrid Basis Functions for Discontinuous Galerkin Schemes on Unstructured Polygonal Voronoi Meshes 下载免费PDF全文
Walter Boscheri Michael Dumbser & Elena Gaburro 《Communications In Computational Physics》2022,32(1):259-298
We propose a new high order accurate nodal discontinuous Galerkin (DG)
method for the solution of nonlinear hyperbolic systems of partial differential equations (PDE) on unstructured polygonal Voronoi meshes. Rather than using classical
polynomials of degree $N$ inside each element, in our new approach the discrete solution
is represented by piecewise continuous polynomials of degree $N$ within each Voronoi element, using a continuous finite element basis defined on a subgrid inside each polygon.
We call the resulting subgrid basis an agglomerated finite element (AFE) basis for the DG
method on general polygons, since it is obtained by the agglomeration of the finite element basis functions associated with the subgrid triangles. The basis functions on each
sub-triangle are defined, as usual, on a universal reference element, hence allowing to
compute universal mass, flux and stiffness matrices for the subgrid triangles once and
for all in a pre-processing stage for the reference element only. Consequently, the construction of an efficient quadrature-free algorithm is possible, despite the unstructured
nature of the computational grid. High order of accuracy in time is achieved thanks
to the ADER approach, making use of an element-local space-time Galerkin finite element predictor.The novel schemes are carefully validated against a set of typical benchmark problems for the compressible Euler and Navier-Stokes equations. The numerical results
have been checked with reference solutions available in literature and also systematically compared, in terms of computational efficiency and accuracy, with those obtained
by the corresponding modal DG version of the scheme. 相似文献
9.
This paper is concerned with a new version of the Osher-Solomon Riemann
solver and is based on a numerical integration of the path-dependent dissipation matrix.
The resulting scheme is much simpler than the original one and is applicable to
general hyperbolic conservation laws, while retaining the attractive features of the original
solver: the method is entropy-satisfying, differentiable and complete in the sense
that it attributes a different numerical viscosity to each characteristic field, in particular
to the intermediate ones, since the full eigenstructure of the underlying hyperbolic system
is used. To illustrate the potential of the proposed scheme we show applications
to the following hyperbolic conservation laws: Euler equations of compressible gasdynamics
with ideal gas and real gas equation of state, classical and relativistic MHD
equations as well as the equations of nonlinear elasticity. To the knowledge of the authors,
apart from the Euler equations with ideal gas, an Osher-type scheme has never
been devised before for any of these complicated PDE systems. Since our new general
Riemann solver can be directly used as a building block of high order finite volume
and discontinuous Galerkin schemes we also show the extension to higher order of
accuracy and multiple space dimensions in the new framework of PNPM schemes on
unstructured meshes recently proposed in [9]. 相似文献
10.
Guo-Quan Shi Huajun Zhu & Zhen-Guo Yan 《Communications In Computational Physics》2022,31(4):1215-1241
A priori subcell limiting approach is developed for high-order flux reconstruction/correction procedure via reconstruction (FR/CPR) methods on two-dimensional unstructured quadrilateral meshes. Firstly, a modified indicator based on
modal energy coefficients is proposed to detect troubled cells, where discontinuities
exist. Then, troubled cells are decomposed into nonuniform subcells and each subcell has one solution point. A second-order finite difference shock-capturing scheme
based on nonuniform nonlinear weighted (NNW) interpolation is constructed to perform the calculation on troubled cells while smooth cells are calculated by the CPR
method. Numerical investigations show that the proposed subcell limiting strategy on
unstructured quadrilateral meshes is robust in shock-capturing. 相似文献
11.
Xiang Lai Zhiqiang Sheng & Guangwei Yuan 《Communications In Computational Physics》2015,18(3):650-672
The extension of diamond scheme for diffusion equation to three dimensions
is presented. The discrete normal flux is constructed by a linear combination of
the directional flux along the line connecting cell-centers and the tangent flux along the
cell-faces. In addition, it treats material discontinuities by a new iterative method. The
stability and first-order convergence of the method are proved on distorted meshes. The
numerical results illustrate that the method appears to be approximate second-order
accuracy for solution. 相似文献
12.
On the Order of Accuracy and Numerical Performance of Two Classes of Finite Volume WENO Schemes 下载免费PDF全文
Rui Zhang Mengping Zhang & Chi-Wang Shu 《Communications In Computational Physics》2011,9(3):807-827
In this paper we consider two commonly used classes of finite volume
weighted essentially non-oscillatory (WENO) schemes in two dimensional Cartesian
meshes. We compare them in terms of accuracy, performance for smooth and shocked
solutions, and efficiency in CPU timing. For linear systems both schemes are high
order accurate, however for nonlinear systems, analysis and numerical simulation results
verify that one of them (Class A) is only second order accurate, while the other
(Class B) is high order accurate. The WENO scheme in Class A is easier to implement
and costs less than that in Class B. Numerical experiments indicate that the resolution
for shocked problems is often comparable for schemes in both classes for the same
building blocks and meshes, despite of the difference in their formal order of accuracy.
The results in this paper may give some guidance in the application of high order finite
volume schemes for simulating shocked flows. 相似文献
13.
A Third Order Conservative Lagrangian Type Scheme on Curvilinear Meshes for the Compressible Euler Equations 下载免费PDF全文
Based on the high order essentially non-oscillatory (ENO) Lagrangian type scheme on quadrilateral meshes presented in our earlier work [3], in this paper we develop a third order conservative Lagrangian type scheme on curvilinear meshes for solving the Euler equations of compressible gas dynamics. The main purpose of this work is to demonstrate our claim in [3] that the accuracy degeneracy phenomenon observed for the high order Lagrangian type scheme is due to the error from the quadrilateral mesh with straight-line edges, which restricts the accuracy of the resulting scheme to at most second order. The accuracy test given in this paper shows that the third order Lagrangian type scheme can actually obtain uniformly third order accuracy even on distorted meshes by using curvilinear meshes. Numerical examples are also presented to verify the performance of the third order scheme on curvilinear meshes in terms of resolution for discontinuities and non-oscillatory properties. 相似文献
14.
Differential Formulation of Discontinuous Galerkin and Related Methods for the Navier-Stokes Equations 下载免费PDF全文
Haiyang Gao Z. J. Wang & H. T. Huynh 《Communications In Computational Physics》2013,13(4):1013-1044
A new approach to high-order accuracy for the numerical solution of conservation laws introduced by Huynh and extended to simplexes by Wang and Gao is renamed CPR (correction procedure or collocation penalty via reconstruction). The CPR
approach employs the differential form of the equation and accounts for the jumps
in flux values at the cell boundaries by a correction procedure. In addition to being
simple and economical, it unifies several existing methods including discontinuous
Galerkin, staggered grid, spectral volume, and spectral difference. To discretize the diffusion terms, we use the BR2 (Bassi and Rebay), interior penalty, compact DG (CDG),
and I-continuous approaches. The first three of these approaches, originally derived
using the integral formulation, were recast here in the CPR framework, whereas the
I-continuous scheme, originally derived for a quadrilateral mesh, was extended to a
triangular mesh. Fourier stability and accuracy analyses for these schemes on quadrilateral and triangular meshes are carried out. Finally, results for the Navier-Stokes
equations are shown to compare the various schemes as well as to demonstrate the
capability of the CPR approach. 相似文献
15.
Mengjiao Jiao Yingda Cheng Yong Liu & Mengping Zhang 《Communications In Computational Physics》2020,28(3):927-966
In this paper, we develop central discontinuous Galerkin (CDG) finite element methods for solving the generalized Korteweg-de Vries (KdV) equations in one
dimension. Unlike traditional discontinuous Galerkin (DG) method, the CDG methods evolve two approximate solutions defined on overlapping cells and thus do not
need numerical fluxes on the cell interfaces. Several CDG schemes are constructed, including the dissipative and non-dissipative versions. L2error estimates are established
for the linear and nonlinear equation using several projections for different parameter
choices. Although we can not provide optimal a priori error estimate, numerical examples show that our scheme attains the optimal (k+1)-th order of accuracy when using
piecewise k-th degree polynomials for many cases. 相似文献
16.
R. Chauvin S. Guisset B. Manach-Perennou & L. Martaud 《Communications In Computational Physics》2022,31(1):293-330
A positivity-preserving, conservative and entropic numerical scheme is presented for the three-temperature grey diffusion radiation hydrodynamics model. More
precisely, the dissipation matrices of the colocalized semi-Lagrangian scheme are defined in order to enforce the entropy production on each species (electron or ion) proportionally to its mass as prescribed in [34]. A reformulation of the model is then considered to enable the derivation of a robust convex combination based scheme. This
yields the positivity-preserving property at each sub-iteration of the algorithm while
the total energy conservation is reached at convergence. Numerous pure hydrodynamics and radiation hydrodynamics test cases are carried out to assess the accuracy of the
method. The question of the stability of the scheme is also addressed. It is observed
that the present numerical method is particularly robust. 相似文献
17.
Yongping Cheng Haiyun Dong Maojun Li & Weizhi Xian 《Communications In Computational Physics》2020,28(4):1437-1463
In this paper, we focus on the numerical simulation of the two-layer shallow water equations over variable bottom topography. Although the existing numerical schemes for the single-layer shallow water equations can be extended to two-layer
shallow water equations, it is not a trivial work due to the complexity of the equations.
To achieve the well-balanced property of the numerical scheme easily, the two-layer
shallow water equations are reformulated into a new form by introducing two auxiliary variables. Since the new equations are only conditionally hyperbolic and their
eigenstructure cannot be easily obtained, we consider the utilization of the central discontinuous Galerkin method which is free of Riemann solvers. By choosing the values
of the auxiliary variables suitably, we can prove that the scheme can exactly preserve
the still-water solution, and thus it is a truly well-balanced scheme. To ensure the
non-negativity of the water depth, a positivity-preserving limiter and a special approximation to the bottom topography are employed. The accuracy and validity of the
numerical method will be illustrated through some numerical tests. 相似文献
18.
Weighted Interior Penalty Method with Semi-Implicit Integration Factor Method for Non-Equilibrium Radiation Diffusion Equation 下载免费PDF全文
Rongpei Zhang Xijun Yu Jiang Zhu Abimael F. D. Loula & Xia Cui 《Communications In Computational Physics》2013,14(5):1287-1303
Weighted interior penalty discontinuous Galerkin method is developed to solve the two-dimensional non-equilibrium radiation diffusion equation on unstructured mesh. There are three weights including the arithmetic, the harmonic, and the geometric weight in the weighted discontinuous Galerkin scheme. For the time discretization, we treat the nonlinear diffusion coefficients explicitly, and apply the semi-implicit integration factor method to the nonlinear ordinary differential equations arising from discontinuous Galerkin spatial discretization. The semi-implicit integration factor method can not only avoid severe time step limits, but also take advantage of the local property of DG methods by which small sized nonlinear algebraic systems are solved element by element with the exact Newton iteration method. Numerical results are presented to demonstrate the validity of discontinuous Galerkin method for high nonlinear and tightly coupled radiation diffusion equation. 相似文献
19.
A Conservative Parallel Iteration Scheme for Nonlinear Diffusion Equations on Unstructured Meshes 下载免费PDF全文
Yunlong Yu Yanzhong Yao Guangwei Yuan & Xingding Chen 《Communications In Computational Physics》2016,20(5):1405-1423
In this paper, a conservative parallel iteration scheme is constructed to solve
nonlinear diffusion equations on unstructured polygonal meshes. The design is based
on two main ingredients: the first is that the parallelized domain decomposition is
embedded into the nonlinear iteration; the second is that prediction and correction
steps are applied at subdomain interfaces in the parallelized domain decomposition
method. A new prediction approach is proposed to obtain an efficient conservative
parallel finite volume scheme. The numerical experiments show that our parallel
scheme is second-order accurate, unconditionally stable, conservative and has linear
parallel speed-up. 相似文献
20.
A High Order Sharp-Interface Method with Local Time Stepping for Compressible Multiphase Flows 下载免费PDF全文
Angela Ferrari Claus-Dieter Munz & Bernhard Weigand 《Communications In Computational Physics》2011,9(1):205-230
In this paper, a new sharp-interface approach to simulate compressible
multiphase flows is proposed. The new scheme consists of a high order WENO finite volume scheme for solving the Euler equations coupled with a high order path-conservative
discontinuous Galerkin finite element scheme to evolve an indicator function
that tracks the material interface. At the interface our method applies ghost cells
to compute the numerical flux, as the ghost fluid method. However, unlike the original
ghost fluid scheme of Fedkiw et al. [15], the state of the ghost fluid is derived
from an approximate-state Riemann solver, similar to the approach proposed in [25],
but based on a much simpler formulation. Our formulation leads only to one single
scalar nonlinear algebraic equation that has to be solved at the interface, instead of
the system used in [25]. Away from the interface, we use the new general Osher-type
flux recently proposed by Dumbser and Toro [13], which is a simple but complete Riemann
solver, applicable to general hyperbolic conservation laws. The time integration
is performed using a fully-discrete one-step scheme, based on the approaches recently
proposed in [5, 7]. This allows us to evolve the system also with time-accurate local
time stepping. Due to the sub-cell resolution and the subsequent more restrictive
time-step constraint of the DG scheme, a local evolution for the indicator function is
applied, which is matched with the finite volume scheme for the solution of the Euler
equations that runs with a larger time step. The use of a locally optimal time step
avoids the introduction of excessive numerical diffusion in the finite volume scheme.
Two different fluids have been used, namely an ideal gas and a weakly compressible
fluid modeled by the Tait equation. Several tests have been computed to assess the
accuracy and the performance of the new high order scheme. A verification of our
algorithm has been carefully carried out using exact solutions as well as a comparison
with other numerical reference solutions. The material interface is resolved sharply
and accurately without spurious oscillations in the pressure field. 相似文献